Question: Simplify and expand the following expression: $ \dfrac{10}{3r + 9}+\dfrac{r + 7}{r - 8} $
In order to add expressions, they must have a common denominator. Get both fractions over a common denominator of $(3r + 9)(r - 8)$ Multiply the first term by $\dfrac{r - 8}{r - 8}$ $ \begin{align*} \dfrac{10}{3r + 9} \times \dfrac{r - 8}{r - 8} & = \dfrac{(10)(r - 8)}{(3r + 9)(r - 8)} \\ & = \dfrac{10r - 80}{(3r + 9)(r - 8)}\end{align*} $ Multiply the second term by $\dfrac{3r + 9}{3r + 9}$ $ \begin{align*} \dfrac{r + 7}{r - 8} \times \dfrac{3r + 9}{3r + 9} & = \dfrac{(r + 7)(3r + 9)}{(r - 8)(3r + 9)} \\ & = \dfrac{3r^2 + 30r + 63}{(r - 8)(3r + 9)}\end{align*} $ Now we have: $ = \dfrac{10r - 80}{(3r + 9)(r - 8)} + \dfrac{3r^2 + 30r + 63}{(r - 8)(3r + 9)} $ Now both terms have a common denominator we can simply add the numerators: $ = \dfrac{10r - 80 + 3r^2 + 30r + 63}{(3r + 9)(r - 8)} $ $ = \dfrac{40r - 17 + 3r^2}{(3r + 9)(r - 8)}$ Expand the denominator: $ = \dfrac{40r - 17 + 3r^2}{3r^2 - 15r - 72}$